Extremal behavior of divisibility functions

Abstract

In this short article, we study the extremal behavior \({{\mathrm{F}}}_\Gamma (n)\) of divisibility functions \({{\mathrm{D}}}_\Gamma \) introduced by the first author for finitely generated groups \(\Gamma \). These functions aim at quantifying residual finiteness and bounds give a measurement of the complexity in verifying a word is non-trivial. We show that finitely generated subgroups of \({{\mathrm{GL}}}(m,K)\) for an infinite field \(K\) have at most polynomial growth for the function \({{\mathrm{F}}}_\Gamma (n)\). Consequently, we obtain a dichotomy for the growth rate of \(\log {{\mathrm{F}}}_\Gamma (n)\) for finitely generated subgroups of \({{\mathrm{GL}}}(n,\mathbf {C})\). We also show that if \({{\mathrm{F}}}_\Gamma (n) \preceq \log \log n\), then \(\Gamma \) is finite. In contrast, when \(\Gamma \) contains an element of infinite order, \(\log n \preceq {{\mathrm{F}}}_\Gamma (n)\). We end with a brief discussion of some geometric motivation for this work.